The Implications of Row Reducing a Square Matrix to the Identity Matrix and its Connection to Matrix Inverses

If square matrix A can be row reduced to the identity, then A must be invertible.

If a square matrix A can be row reduced to the identity matrix, then it implies that all of its rows are linearly independent and that it spans the entire vector space

If a square matrix A can be row reduced to the identity matrix, then it implies that all of its rows are linearly independent and that it spans the entire vector space. In other words, the matrix A can transform any vector in its vector space into the corresponding vector in the standard coordinate system.

Now, if A is invertible, it means that there exists another matrix B such that the product of A and B is the identity matrix. In other words, AB = BA = I, where I is the identity matrix.

To prove that A is invertible if it can be row reduced to the identity, we can use the fact that row operations preserve the row space and the null space of the matrix.

When A is row reduced to the identity matrix, it means that all the rows are linearly independent, and there are no rows consisting entirely of zeros. This ensures that the null space of A is only the trivial solution, which is {0}, that is, the only solution to the equation Ax = 0 is x = 0.

Since the null space of A contains only the zero vector, it means that A has a pivot position in every column when row reduced to the identity matrix. Consequently, the equation Ax = b has a unique solution for every vector b.

Now, let’s assume that A is not invertible, which means that there exists a nonzero vector b for which the equation Ax = b has no solution. However, this contradicts the fact that A, when row reduced to the identity matrix, can map any vector in its vector space to the standard coordinate system.

Hence, our assumption that A is not invertible leads to a contradiction. Therefore, if a square matrix A can be row reduced to the identity matrix, A must be invertible.

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